In the simplest 1 D case, let \(\mathbf {x}\)  be the input tensor, \(\mathbf {w}\)  be the convolutional filter, and \(c\)  be the support of the convolution (i.e. the set of indices specifying which elements of \(x\)  should be used in the convolution). The output of the sparse convolution, \(\mathbf {y}\) , can be computed as: \(\mathbf {y} = \mathbf {x}[c] * \mathbf {w}\) 
where \(*\)  denotes the convolution operation, and \(\mathbf {x}[c]\)  is the subset of elements from \(\mathbf {x}\)  specified by the support \(c\) . This equation applies the convolutional filter \(\mathbf {w}\)  to the selected input elements and sums the results to produce the output of the convolution. For more detailed formulation and implementation of the Sparse convolutions we used in our work, please refer to MinkowskiNetwork [1]}.
